Uniform bounds in D-minimal structures

Farris, Madie

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https://hdl.handle.net/2142/129922 Copy

Description

Title

Uniform bounds in D-minimal structures

Author(s)

Farris, Madie

Issue Date

2025-07-08

Director of Research (if dissertation) or Advisor (if thesis)

Hieronymi, Philipp

Doctoral Committee Chair(s)

van den Dries, Lou

Committee Member(s)

• Castle, Ben
• Miller, Chris

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

d-minimality, model theory, o-minimality

Abstract

O-minimality as a classification of the topological tameness of a structure has been extensively studied since its introduction in 1982. Many generalizations and variations of o-minimality have since been introduced and studied as well. In this thesis we focus on one particular generalization: d-minimality. We establish some of the first results on d-minimality that truly mirror those of o-minimality. The most central of which is the equivalence of d-minimality and strong d-minimality. In the process of proving this equivalence we develop a d-minimal version of cell decomposition, one of the most important results in o-minimality. In Chapter 1 we discuss the overarching project of tame topology, and situate d-minimality within this context. We define a grid, the object that we will use to decompose sets in d-minimal structures, and our main results, including the aforementioned decomposition theorem. Chapter 2 is spent establishing some preliminary facts that are used throughout this thesis. In Chapter 3 we motivate our choice of definition for grids by working through an example of a grid decomposition, and compare this decomposition to a previously known result for strongly d-minimal structures. We develop the theory of grids (and cells and stacks which are used to build grids) in Chapter 4. We put all of these pieces together in Chapter 5 where we prove our main results. In Chapter 6 we extend these results to a more general setting. Lastly, we discuss future applications of our main results in Chapter 7.

Graduation Semester

2025-08

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/129922

Copyright and License Information

Copyright 2025 Madie Farris

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